where \(\mathbf {x}\)  is a voxel and \(c\)  is a class. \(p_c(\mathbf {x})\)  is the predicted probability of being class \(c\)  for voxel \(\mathbf {x}\) , and \(g_c(\mathbf {x})\)  is the corresponding ground truth label. \(\mathbf {E[\cdot ]}\)  is the mean value with respect to \(c\)  and \(\mathbf {x}\)  in \(L_{DSC}\)  and \(L_{Cross}\) , respectively. \(\epsilon =1\)  is the pseudocount for additive smoothing to handle missing labels in training samples. \(w_c = ((\sum _k f_k)/f_c)^{0.5}\)  is the class-level weight for reducing the influence of more frequently seen classes, where \(f_k\)  is the frequency of class \(k\) . We set \(\omega _{DSC} , \omega _{Cross}, \gamma _{DSC}, \gamma _{Cross}=1\)  in our experiments according to [1]}.
